Topological entropy

Topological entropy video

In dynamical systems, complexity is usually measured by the topological entropy and reflects roughly speaking, the proliferation of periodic orbits with ever longer periods or the number of orbits that can be distinguished with increasing precision.

See the related Kolmogorov-Sinai entropy

Hans Henrik RUGH - The Milnor-Thurston determinant and the Ruelle transfer operator

Descriptional complexity

For a coarse-grained Dynamical system, described by a transition graph, in turn described by an Adjacency matrix $A$, then the topological entropy $h$ is

$h = \log{\lambda_{\text{max}}}$

where $\lambda_{\text{max}}$ is the maximum eigenvalue of $A$ (assumed to be a positive matrix so that Perron-Frobenius applies).

Determinant of a graph